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M8130 Algebraic topology, tutorial 10, 2020 16. 12. 2020
Exercise 1. Show -nk(S°°) = 0 for all k, where S00 is colimSn. Using it show that S00 is homotopy equivalent to a point.
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M8130 Algebraic topology, tutorial 10, 2020 16. 12. 2020
Exercise 2. Compute homotopy groups 0/BLP00.
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M8130 Algebraic topology, tutorial 10, 2020_16. 12. 2020
Exercise 3. Show that the spaces S2 x RP°° and RP2 have the same homotopy groups but they are not homotopy equivalent.
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M8130 Algebraic topology, tutorial 10, 2020
16. 12. 2020
Exercise 4. Extension lemma: Let (X, A) be a pair of CW-complexes, Y a space with ^n-iiXiVo) = 0 whenever there is a cell of dimension n in X — A. Then every map f:A^-Y can be extended to a map F: X -^Y.
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M8130 Algebraic topology, tutorial 10, 2020 16. 12. 2020
Exercise 5. Long exact sequence of the fibration (Hopf) S1 —> S3 —> CP1 = S2.
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M8130 Algebraic topology, tutorial 10, 2020
16. 12. 2020
Exercise 6. Using homotopy groups show that M.Pk is not a retract ofM.Pn, n > k > 1.
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M8130 Algebraic topology, tutorial 10, 2020
16. 12. 2020
Exercise 7. Consider the map q: S1xS1xS1 —> S3 defined as a map S1 x S1 x S1 —> D3/S2 where D3 is a small disk in the triple torus which is the identity in the interior of D3 and constant on its complement. Further, consider the Hopf map p: S3 —>• S2 = CP1 (described in Hopf fibration S1 —> S3 —> S2). Compute g* and (pq)* in homotopy groups. Show that pq is not homotopic to a constant map. ^ ^
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